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Birkhäuser - Birkhäuser Mathematics | A Course on Integration Theory - including more than 150 exercises with detailed answers

A Course on Integration Theory

including more than 150 exercises with detailed answers

Lerner, Nicolas

2014, XVIII, 492 p. 15 illus., 3 illus. in color.

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  • Includes over 150 exercises with detailed solutions
  • Requires no prior knowledge of advanced mathematics although the results proven in the book are not elementary
  • Is self-contained, providing detailed arguments for each statement
  • Includes a helpful appendix to recall basic notions

This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the Riesz-Markov Theorem and also via the Carathéodory Theorem. It also includes some elementary properties of Hausdorff measures as well as the basic properties of spaces of integrable functions and standard theorems on integrals depending on a parameter. Integration on a product space, change-of-variables formulas as well as the construction and study of classical Cantor sets are treated in detail. Classical convolution inequalities, such as Young's inequality and Hardy-Littlewood-Sobolev inequality, are proven. Further topics include the Radon-Nikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems including Marcinkiewicz's theorem, and the definition of Lebesgue points and the Lebesgue differentiation theorem.

Each chapter ends with a large number of exercises and detailed solutions.

A comprehensive appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. It also provides more advanced material such as some basic properties of cardinals and ordinals which are useful for the study of measurability.

Content Level » Graduate

Keywords » Fourier transformation - L^p spaces - Lebesgue measure - measure theory

Related subjects » Birkhäuser Mathematics

Table of contents 

1 Introduction.- 2 General theory of integration.- 3 Construction of the Lebesgue measure on R^d.- 4 Spaces of integrable functions.- 5 Integration on a product space.- 6 Diffeomorphisms of open subsets of R^d and integration.- 7 Convolution.- 8 Complex measures.- 9 Harmonic analysis.- 10 Classical inequalities.

Distribution rights 

Distribution rights for India: Researchco Book Centre, New Delhi, India

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